The inner graph \(H\) . The choice of \(H\)  is
similar to Ta-Shma's choice. More precisely, we set \(s=1/\alpha \)  and
\(d_2 = s^{4s^2}\)  (Ta-Shma took \(d_2 = s^{4s}\) ).
We obtain a Cayley graph \(H= \textup {Cay}(\mathbb {F}_2^{4s\log _2(d_2)}, A)\)  such that \(H\)  is an
\((n_2=d_2^{4s}, d_2, \lambda _2)\)  graph where
\(\lambda _2 = b_2/\sqrt{d_2}\)  and \(b_2 = 4 s \log _2(d_2)\) .
(The set of generators, \(A\) , comes from a small bias code derived from
a construction of Alon et al. [1]}, but we will rely on
Ta-Shma's analysis embodied in lemma:aghp and not discuss it
further.)
